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1.a) CD4 count changes with age CD4 Count drops with age in healthy children Huenecke et al. [1] 3

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1.a) CD4 count changes with age This show the raw data plotted against age, with the Heunecke et al. [1] curve for healthy children As can be seen, this change with age is clear in the data 4

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1.b) Accounting for this change Pre-adjusting for change Log of the ratio between the measured CD4 and the expected for a healthy child Square root of this ratio Fourth root of this ratio Building it into the model Bi-linear model for asymptote with age Exponential decay model for asymptote Asymptote fixed as Heunecke curve Asymptote as multiple of Heunecke curve No consistency in the results Can we know which is the correct method to model this change with age? 5

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2. Issues with the basic model Still need to find a way to account for age Tried many models for age adjustment –Log ratio –Square root ratio –4 th root ratio –Age adjustment of parameters themselves Also tried letting model chose best form of age adjustment through SCM Still no consistency 8

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3.a) A new way of applying the model More sensible to model the total CD4 cells, rather than the CD4 concentration λ in the model becomes thymic output in cells per day δ the net rate of loss of cells in units of per day We can then use information about changes to the thymus and to the body Parameters are then more biologically relevant 10

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3.b) Converting concentration to total Do this as per Bains et al. [2] Use the WHO data from Kuczmarski et al. [3] to calculate weight for age Then use equation from Linderkamp et al. [4] to find volume from weight 11

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4. The updated model Same turnover model as before But functional parameters chosen such that they reflect the underlying system Form for thymic output chosen from Bains et al. [5] where: With y(t) the proportion of cells expresing Ki67, v(t) the Heunecke et al. [1] relation for CD4 concentration with age, V(t) the blood volume with time found from the weight with time Peripheral CD4 cells (X) λD P 15

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4. The updated model Form for net rate of cell loss chosen such that the form of the total CD4 with age matches the expected total body CD4 with θ λ = θ δ = 1 With y(t) defined as before. Below shows the comparison with θ λ = θ δ = 1. 16

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4. Issues with the updated model NONMEM finds the best fit with a very low value for δ and λ, corresponding to 0.1 to 0.01 times the expected values for δ and λ. The plots are not a good representation of the data. 17

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4. Issues with the updated model Low value for δ is caused by the slow increase in CD4 after the BMT Low value for λ then required to keep a sensible asymptote Two options to slow the increase in total CD4 immediately after BMT -Decrease δ for low times, slowing the dynamics of the system -Decrease λ for low times, lowering the driving force for the increase 18

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5. Thymic effects Evidence from TREC analysis suggests that the thymus does not recover full functionality for between 6 and 12 months after a BMT To achieve this initial decrease in thymic function after the BMT, I used a logistic regression on λ: Where θ 1 and θ 2 are new parameters to be estimated θ 1 corresponds to the rate of increase of the value of λ θ 2 corresponds to the time it takes for λ or δ to reach half its eventual value On λ, it causes a drop in objective function of 419. 20

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5. Competition effects Other evidence suggests that competition for homeostatic signals, such as IL- 7 cytokines, may affect proliferation and loss rates for CD4 cells This effect on net loss can be modelled as: Where X is the total number of CD4 cells and θ half is the value of X at which δ will reach half its final value. At the moment I fix θ half to 60,000 cells This causes a further drop in objective function of 257 So the combination of the two give a drop of 676 21

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5. Visual predictive check plot The VPC seems to model the data well, with nearly all points for the data percentiles falling within the model prediction bands. The high 97.5 th percentile band could be due to truncated data 24

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5. Model compared with expected A plot of the theoretical progression of a child having a BMT at age 0 days old, against the expected progression of a health child, demonstrating the early delay followed by the reconstitution in a very similar shape 25

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5. Problems with the model and further work We have two competing functions for the initial stages after the BMT There is certain to be some inter-dependence between the parameters, particularly between the three for the thymic and competition effects Therefore need to be careful for this interdependence, and hence the value for θ 3 has been fixed for now Some work needs to be done on the competition function – there needs to be some allowance for the fact that not every patient is heading for the same asymptote due to inter-individual differences in λ and δ and fixed age effects Then onto multivariate analysis with Lasso and SCM 27

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6. Conclusions We now have a good model for immune reconstitution of children post BMT that has good predictive qualities We have found fixed effects to account for the changes in the immune system with age Including the effects of reduced thymic production is very important Including the effects of competition for homeostatic signals also important However some issues with the model still need to be sorted We can then proceed to multivariate analysis 29